Lectures on Holomorphic Curves in Symplectic and Contact Geometry

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Lectures on Holomorphic Curves in Symplectic and Contact Geometry

(Version 2.0)

Chris Wendl

Institut f¨ur Mathematik, Humboldt-Universit¨at zu Berlin E-mail address: wendl@math.hu-berlin.de

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°2009-2010 by Chris Wendlc

Paper or electronic copies for noncommercial use may be made freely without explicit permission from the author. All other rights reserved.

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Introduction

Ifm= 1, so U is an open subset of C, we refer to holomorphic mapsu:U →Cⁿ as holomorphic curves in Cⁿ. The choice of wording is slightly unfortunate if you like to think in terms of real geometry—after all, the image of u looks more like a surface than a curve. But we call u a “curve” because, in complex terms, it is a one-dimensional object.

That said, let us think of holomorphic curves for the moment as real 2-dimensional objects and ask a distinctly real 2-dimensional question: what is the area traced out by u : U → Cⁿ? Denote points in U by s+it and think of u as a function of the two real variables (s, t), with values in R²ⁿ. In these coordinates, the action of i on vectors in C=R² can be expressed succinctly by the relation

i∂s =∂t.

We first have to compute the area of the parallelogram in R²ⁿ spanned by ∂su(s, t) and ∂tu(s, t). The Cauchy-Riemann equation (1.1) makes this easy, because

∂tu(s, t) = du(s, t)∂t=du(s, t)i∂s=i du(s, t)∂s =i ∂su(s, t),

which implies that ∂su(s, t) and∂tu(s, t) are orthogonal vectors of the same length.

Thus the area of u is Area(u) =

Z

U

|∂su||∂tu| ds dt= 1 2

Z

U

¡|∂su|²+|∂tu|²¢

ds dt,

where we’ve used the fact that |∂su|= |∂tu| to write things slightly more symmet- rically. Notice that the right hand side is really an analytical quantity: up to a constant it is the square of the L²-norm of the first derivative of u.

Let us now write this area in a slightly different, more topological way. If h , i denotes the standard Hermitian inner product on Cⁿ, notice that one can define a differential 2-form on R²ⁿ by the expression

ω₀(X, Y) = RehiX, Yi.

Writing points inCⁿ via the coordinates (p1+iq1, . . . , pn+iqn), one can show that ω0 in these coordinates takes the form

(1.2) ω₀ =

n

X

j=1

dpj ∧dqj.

Exercise 1.2. Prove (1.2), and then show that ω₀ has the following three properties:

(1) It is nondegenerate: ω0(V,·) = 0 for some vector V if and only if V = 0.

Equivalently, for each z ∈ R²ⁿ, the map TzR²ⁿ → T_z^∗R²ⁿ :V 7→ ω0(V,·) is an isomorphism.

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(2) It is closed: dω0 = 0.

(3) The n-fold product ω₀ⁿ=ω0∧. . .∧ω0 is a constant multiple of the natural volume form on R²ⁿ.

Exercise 1.3. Show that a 2-form ω onR²ⁿ (and hence on any 2n-dimensional manifold) is nondegenerate if and only if ωⁿ is a volume form.

Using ω0, we see that the area of the parallelogram above is also

|∂su| · |∂tu|=|∂tu|² = Reh∂tu, ∂tui= Rehi∂su, ∂tui=ω0(∂su, ∂tu), thus

(1.3) Area(u) =kduk²_L2 =

Z

U

u^∗ω0.

This is the first appearance of symplectic geometry in our study of holomorphic curves; we call ω0 the standard symplectic form on R²ⁿ. The point is that the expression on the right hand side of (1.3) is essentially topological: it depends only on the evaluation of a certain closed 2-form on the 2-chain defined by u(U). The present example is trivial because we’re only working in R²ⁿ, but as we’ll see later in more interesting examples, one can often find an easy topological bound on this integral, which by (1.3) implies a bound on the analytical quantitykduk²_L2. One can use this to derive compactness results for spaces of holomorphic curves, which then encode symplectic topological information about the space in which these curves live. We’ll come back to this theme again and again.

2. Hamiltonian systems and symplectic manifolds

To motivate the study of symplectic manifolds in general, let us see how symplectic structures arise naturally in classical mechanics. We shall only sketch the main ideas here; a good comprehensive introduction may be found in [Arn89].

Consider a mechanical system with “n degrees of freedom” moving under the influence of a Newtonian potential V. This means there are n “position” variables q = (q1, . . . , qn) ∈ Rⁿ, which are functions of time t that satisfy the second order differential equation

(1.4) miq¨i =−∂V

∂qi

,

where mi > 0 are constants representing the masses of the various particles, and V : Rⁿ → R is a smooth function, the “potential”. The space Rⁿ, through which the vectorq(t) moves, is called theconfiguration space of the system. The basic idea of Hamiltonian mechanics is to turn this 2nd order system into a 1st order system by introducing an extra set of “momentum” variables p= (p1, . . . , pn)∈Rⁿ, where pi =miq˙i. The space R²ⁿ with coordinates (p, q) is then called phase space, and we define a real valued function on phase space called the Hamiltonian, by

H :R²ⁿ →R: (p, q)7→ 1 2

n

X

i=1

p²_i mi

+V(q).

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Physicists will recognize this as the “total energy” of the system, but its main significance in the present context is that the combination of the second order system (1.4) with our definition ofp is now equivalent to the 2n first order equations,

(1.5) q˙i = ∂H

∂pi

, p˙i =−∂H

∂qi

. These are Hamilton’s equations for motion in phase space.

The motion of x(t) := (p(t), q(t)) in R²ⁿ can be described in more geometric terms: it is an orbit of the vector field

(1.6) XH(p, q) =

n

X

i=1

µ∂H

∂pi

∂

∂qi

−∂H

∂qi

∂

∂pi

¶ .

As we’ll see in a moment, vector fields of this form have some important properties that have nothing to do with our particular choice of the function H, thus it is sensible to call any vector field defined by this formula (for an arbitrary smooth function H : R²ⁿ → R) a Hamiltonian vector field. This is where the symplectic structure enters the story.

Exercise 1.4. Show that the vector field XH of (1.6) can be characterized as the unique vector field on R²ⁿ that satisfies ω0(XH,·) = −dH.

The above exercise shows that the symplectic structure makes it possible to write down a much simplified definition of the Hamiltonian vector field. Now we can already prove something slightly impressive.

Proposition 1.5. The flow ϕ^t_H of XH satisfies (ϕ^t_H)^∗ω0 =ω0 for all t.

Proof. Using Cartan’s formula for the Lie derivative of a form, together with the characterization ofXH in Exercise1.4and the fact thatω0 is closed, we compute LXHω0 =dιXHω0+ιXHdω0 =−d²H = 0. ¤ By Exercise 1.2, one can compute volumes on R²ⁿ by integrating the n-fold product ω0∧. . .∧ω0, thus an immediate consequence of Prop. 1.5 is the following:

Corollary 1.6 (Liouville’s theorem). The flow of XH is volume preserving.

Notice that in most of this discussion we’ve not used our precise knowledge of the 2-form ω0 or function H. Rather, we’ve used the fact that ω0 is nondegenerate (to characterizeXH viaω0 in Exercise1.4), and the fact that it’s closed (in the proof of Prop. 1.5). It is therefore natural to generalize as follows.

Definitions 1.7. A symplectic form on a 2n-dimensional manifold M is a smooth differential 2-form ω that is both closed and nondegenerate. The pair (M, ω) is then called a symplectic manifold. Given a smooth function H :M →R, the corresponding Hamiltonian vector field is defined to be the unique vector field XH ∈Vec(M) such that¹

(1.7) ω(XH,·) = −dH.

1Some sources in the literature defineXH byω(XH,·) =dH, in which case one must choose different sign conventions for the orientation of phase space and definition ofω0. One must always be careful not to mix sign conventions from different sources—that way you could prove anything!

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For two symplectic manifolds (M1, ω1) and (M2, ω2), a smooth map ϕ : M1 → M2

is called symplectic if ϕ^∗ω2 = ω1. If ϕ is a symplectic embedding, then we say that ϕ(M1) is a symplectic submanifold of (M2, ω2). If ϕ is symplectic and is also a diffeomorphism, it is called a symplectomorphism, and we then say that (M1, ω1) and (M2, ω2) are symplectomorphic.

Repeating verbatim the argument of Prop.1.5, we see now that any Hamiltonian vector field on a symplectic manifold (M, ω) defines a smooth 1-parameter family of symplectomorphisms. If we define volumes onM by integrating the 2n-form ωⁿ (see Exercise 1.3), then all symplectomorphisms are volume preserving—in particular this applies to the flow of XH.

Remark 1.8. An odd-dimensional manifold can never admit a nondegenerate 2-form. (Why not?)

3. Some favorite examples

We now give a few examples of symplectic manifolds (other than (R²ⁿ, ω0)) which will be useful to have in mind.

Example1.9. SupposeN is any smoothn-manifold and (q1, . . . , qn) are a choice of coordinates on an open subset U ⊂ N. These naturally define coordinates (p1, . . . , pn, q1, . . . , qn) on the cotangent bundle T^∗U ⊂ T^∗N, where an arbitrary cotangent vector at q∈ U is expressed as

p1 dq1+. . .+pn dqn.

Interpreted differently, this expression also defines a smooth 1-form on T^∗U; we abbreviate it by p dq.

Exercise1.10. Show that the 1-formp dqdoesn’t actually depend on the choice of coordinates (q1, . . . , qn).

What the above exercise reveals is that T^∗N globally admits a canonical 1-form λ, whose expression in the local coordinates (p, q) always looks like p dq. Moreover, dλ is clearly a symplectic form, as it looks exactly like (1.2) in coordinates. We call this the canonical symplectic form on T^∗N. Using this symplectic structure, the cotangent bundle can be thought of as the “phase space” of a smooth manifold, and is a natural setting for studying Hamiltonian systems when the configuration space is something other than a Euclidean vector space (e.g. a “constrained” mechanical system).

Example 1.11. On any oriented surface Σ, a 2-form ω is symplectic if and only if it is an area form, and the symplectomorphisms are precisely the area preserving diffeomorphisms. Observe that one can always find area preserving diffeomorphisms between small open subsets of (R², ω0) and (Σ, ω), thus every point in Σ has a neighborhood admitting local coordinates (p, q) in which ω=dp∧dq.

Example 1.12. A more interesting example of a closed symplectic manifold is then-dimensional complex projective spaceCPⁿ. This is both a real 2n-dimensional symplectic manifold and a complex n-dimensional manifold, as we will now show.

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By definition, CPⁿ is the space of complex lines in Cⁿ⁺¹, which we can express in two equivalent ways as follows:

CPⁿ= (Cⁿ⁺¹\ {0})/C^∗ =S²ⁿ⁺¹/S¹.

In the first case, we divide out the natural free action (by scalar multiplication) of the multiplicative group C^∗ := C\ {0} on Cⁿ⁺¹ \ {0}, and the second case is the same thing but restricting to the unit sphere S²ⁿ⁺¹ ⊂Cⁿ⁺¹ =R²ⁿ⁺² and unit circle S¹ ⊂ C = R². To define a symplectic form, consider first the 1-form λ on S²ⁿ⁺¹ defined for z∈S²ⁿ⁺¹ ⊂Cⁿ⁺¹ and X ∈TzS²ⁿ⁺¹ ⊂Cⁿ⁺¹ by

λz(X) = hiz, Xi,

where h , i is the standard Hermitian inner product on Cⁿ⁺¹. (Take a moment to convince yourself that this expression is always real.) Since λ is clearly invariant under the S¹-action on S²ⁿ⁺¹, the same is true for the closed 2-form dλ, which therefore descends to a closed 2-form ω0 on CPⁿ.

Exercise 1.13. Show that ω0 as defined above is symplectic.

The complex manifold structure of CPⁿ can be seen explicitly by thinking of points in CPⁿ as equivalence classes of vectors (z0, . . . , zn) ∈ Cⁿ⁺¹\ {0}, with two vectors equivalent if they are complex multiples of each other. We will always write the equivalence class represented by (z₀, . . . , zn)∈Cⁿ⁺¹\ {0} as

[z0 :. . .:zn]∈CPⁿ. Then for each k = 0, . . . , n, there is an embedding

(1.8) ιk :Cⁿ֒→CPⁿ: (z1, . . . , zn)7→[z1 :. . . , zk−1 : 1 : zk :. . .:zn], whose image is the complement of the subset

CPⁿ⁻¹ ∼={[z1 :. . .:zk−1 : 0 :zk :. . .:zn]∈CPⁿ |(z1, . . . , zn)∈Cⁿ}.

Exercise 1.14. Show that if the mapsι⁻_k¹ are thought of as complex coordinate charts on open subsets ofCPⁿ, then the transition mapsι⁻_k¹◦ιj are all holomorphic.

By the exercise, CPⁿ naturally carries the structure of a complex manifold such that the embeddings ιk : Cⁿ → CPⁿ are holomorphic. Each of these embeddings also defines a decomposition of CPⁿ into Cⁿ∪CPⁿ⁻¹, where CPⁿ⁻¹ is a complex submanifold of (complex) codimension one. The case n = 1 is particularly enlight- ening, as here the decomposition becomes CP¹ =C∪ {point} ∼= S²; this is simply the Riemann sphere with its natural complex structure, where the “point at infinity”

is CP⁰. In the case n = 2, we haveCP² ∼=C²∪CP¹, and we’ll occasionally refer to the complex submanifold CP¹ ⊂CP² as the “sphere at infinity”.

We continue for a moment with the example of CPⁿ in order to observe that it contains an abundance of holomorphic spheres. Take for instance the case n = 2:

then for any ζ ∈C, we claim that the holomorphic embedding uζ :C→C² :z 7→(z, ζ)

extends naturally to a holomorphic embedding of CP¹ in CP². Indeed, using ι2

to include C² in CP², uζ(z) becomes the point [z : ζ : 1] = [1 : ζ/z : 1/z], and

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CP²

CP¹

x0

Figure 1. CP² \ {x₀} is foliated by holomorphic spheres that all intersect at x₀.

as z → ∞, this converges to the point x0 := [1 : 0 : 0] in the sphere at infinity.

One can check using alternate charts that this extension is indeed a holomorphic map. The collection of all these embeddingsuζ :CP¹ →CP² thus gives a very nice decomposition of CP²: together with the sphere at infinity, they foliate the region CP²\ {x₀}, but all intersect precisely at x₀ (see Figure 1). This decomposition will turn out to be crucial in the proof of Theorem 1.25, stated below.

4. Darboux’s theorem and the Moser deformation trick

In Riemannian geometry, two Riemannian manifolds of the same dimension with different metrics can have quite different local structures: there can be no isometries between them, not even locally, unless they have the same curvature. The following basic result of symplectic geometry shows that in the symplectic world, things are quite different. We will give a proof using the beautiful Moser deformation trick, which has several important applications throughout symplectic and contact geometry, as we’ll soon see.²

Theorem 1.15 (Darboux’s theorem). Near every point in a symplectic manifold (M, ω), there are local coordinates (p1, . . . , pn, q1, . . . , qn)in which ω=P

idpi∧dqi. Proof. Denote by (p1, . . . , pn, q1, . . . , qn) the standard coordinates on R²ⁿ and define the standard symplectic formω0 by (1.2); this is the exterior derivative of the 1-form

λ0 =X

j

pj dqj.

Since the statement in the theorem is purely local, we can assume (by choosing local coordinates) thatM is an open neighborhood of the origin inR²ⁿ, on whichωis any

2An alternative approach to Darboux’s theorem may be found in [Arn89].

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closed, nondegenerate 2-form. Then it will suffice to find two open neighborhoods U,U0 ⊂R²ⁿ of 0, and a diffeomorphism

ϕ :U0 → U

preserving 0 such that ϕ^∗ω=ω0. Using Exercise 1.16below (the “linear Darboux’s theorem”), we can also assume after a linear change of coordinates that ϕ^∗ωand ω0

match at the origin.

The idea behind the Moser trick is now the following bit of optimism: we assume that the desired diffeomorphism ϕis the time 1 flow of a time-dependent vector field defined near 0, and derive conditions that this vector field must satisfy. In fact, we will be a bit more ambitious: consider the smooth 1-parameter family of 2-forms

ωt=tω+ (1−t)ω0, t∈[0,1]

which interpolate between ω0 and ω. These are all obviously closed, and if we restrict to a sufficiently small neighborhood of the origin then they are near ω0 and thus nondegenerate. Our goal is to find a time-dependent vector field Yt on some neighborhood of 0, for t ∈ [0,1], whose flow ϕt is well defined on some smaller neighborhood of 0 and satisfies

ϕ^∗_tωt =ω₀

for all t ∈ [0,1]. Differentiating this expression with respect to t and writing ˙ωt :=

∂

∂tωt, we find

ϕ^∗_tLYtωt+ϕ^∗_tω˙t = 0,

which by Cartan’s formula and the fact thatωtis closed andϕtis a diffeomorphism, implies

(1.9) dιYtωt+ ˙ωt = 0.

At this point it’s useful to observe that if we restrict to a contractible neighborhood of the origin, ω (and hence also ωt) is exact: let us write

ω =dλ.

Moreover, by adding a constant 1-form, we can choose λ so that it matches λ0 at the origin. Now if λt :=tλ+ (1−t)λ0, we have dλt =ωt, and ˙λt := _∂t^∂λt =λ−λ0

vanishes at the origin. Plugging this into (1.9), we see now that it suffices to find a vector field Yt satisfying

(1.10) ωt(Yt,·) = −λ˙t.

Since ωt is nondegenerate, this equation can be solved and determines a unique vector field Yt, which vanishes at the origin since ˙λt does. The flow ϕt therefore exists for all t ∈ [0,1] on a sufficiently small neighborhood of the origin, and ϕ1 is

the desired diffeomorphism. ¤

Exercise 1.16. The following linear version of Darboux’s theorem is an easy exercise in linear algebra and was the first step in the proof above: show that if Ω is any nondegenerate, antisymmetric bilinear form on R²ⁿ, then there exists a basis (X1, . . . , Xn, Y1, . . . , Yn) such that

Ω(Xi, Yi) = 1

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and Ω vanishes on all other pairs of basis vectors. This is equivalent to the statement that R²ⁿ admits a linear change of coordinates in which Ω looks like the standard symplectic form ω0.

It’s worth pointing out the crucial role played in the above proof by the relation (1.10), which is almost the same as the relation used to define Hamiltonian vector fields (1.7). The latter, together with the argument of Prop. 1.5, tells us that the group of symplectomorphisms on a symplectic manifold is fantastically large, as it contains all the flows of Hamiltonian vector fields, which are determined by arbitrary smooth real valued functions. For much the same reason, one can also always find an abundance of symplectic local coordinate charts (usually called Darboux co- ordinates). Contrast this with the situation on a Riemannian manifold, where the group of isometries is generally finite dimensional, and different metrics are usually not locally equivalent, but are distinguished by their curvature.

In light of Darboux’s theorem, we can now give the following equivalent definition of a symplectic manifold:

Definition 1.17. A symplectic manifold is a 2n-dimensional manifold M together with an atlas of coordinate charts whose transition maps are symplectic (with respect to the standard symplectic structure of R²ⁿ).

In physicists’ language, a symplectic manifold is thus a manifold that can be identified locally with Hamiltonian phase space, in the sense that all coordinate changes leave the form of Hamilton’s equations unaltered.

Let us state one more important application of the Moser trick, this time of a more global nature. Recall that two symplectic manifolds (M, ω) and (M^′, ω^′) are called symplectomorphic if there exists a symplectomorphism between them, i.e. a diffeomorphism ϕ : M → M^′ such that ϕ^∗ω^′ = ω. Working on a single manifold M, we say similarly that two symplectic structures ω and ω^′ are symplectomorphic³ if (M, ω) and (M, ω^′) are symplectomorphic. This is the most obvious notion of equivalence for symplectic structures, but there are others that are also worth considering.

Definition 1.18. Two symplectic structuresω and ω^′ onM are calledisotopic if there is a symplectomorphism (M, ω)→(M, ω^′) that is isotopic to the identity.

Definition1.19. Two symplectic structuresωandω^′ onM are calleddeforma- tion equivalent if M admits a symplectic deformation between them, i.e. a smooth family of symplectic forms {ωt}t∈[0,1] such that ω0 =ω and ω1 =ω^′. Similarly, two symplectic manifolds (M, ω) and (M^′, ω^′) are deformation equivalent if there exists a diffeomorphism ϕ :M →M^′ such that ω and ϕ^∗ω^′ are deformation equivalent.

It is clear that if two symplectic forms are isotopic then they are also both symplectomorphic and deformation equivalent. It is not true, however, that a symplectic deformation always gives rise to an isotopy: one should not expect this, as isotopic symplectic forms onM must always represent the same cohomology class inH_dR² (M), whereas the cohomology class can obviously vary under general deformations. The remarkable fact is that this necessary condition is also sufficient!

3The words “isomorphic” and “diffeomorphic” can also be used here as synonyms.

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Theorem 1.20 (Moser’s stability theorem). Suppose M is a closed manifold with a smooth 1-parameter family of symplectic forms {ωt}[t∈[0,1] which all represent the same cohomology class in H_dR² (M). Then there exists a smooth isotopy {ϕt : M →M}t∈[0,1], with ϕ0 = Id and ϕ^∗_tωt=ω0.

Exercise 1.21. Use the Moser isotopy trick to prove the theorem. Hint: in the proof of Darboux’s theorem, we had to use the fact that symplectic forms are locally exact in order to get from(1.9)to(1.10). Here you will find the cohomological hypothesis helpful for the same reason. If you get stuck, see [MS98].

Exercise1.22. Show that ifωandω^′are two deformation equivalent symplectic forms on CPⁿ, then ω is isotopic to cω^′ for some constant c >0.

5. From symplectic geometry to symplectic topology

As a consequence of Darboux’s theorem, symplectic manifolds have no local invariants—there is no “local symplectic geometry”. Globally things are different, and here there are a number of interesting questions one can ask, all of which fall under the heading of symplectic topology. (The word “topology” is used to indicate the importance of global rather than local phenomena.)

The most basic such question concerns the classification of symplectic structures.

One can ask, for example, whether there exists a symplectic manifold (M, ω) that is diffeomorphic to R⁴ but not symplectomorphic to (R⁴, ω0), i.e. an “exotic” symplectic R⁴. The answer turns out to be yes—exotic R²ⁿ’s exist in fact for all n, see [ALP94]—but it changes if we prescribe the behavior ofω at infinity. The following result says that (R²ⁿ, ω0) is actually the only aspherical symplectic manifold that is

“standard at infinity”.

Theorem 1.23 (Gromov [Gro85]). Suppose (M, ω) is a symplectic 4-manifold with π2(M) = 0, and there are compact subsets K ⊂ M and Ω ⊂ R⁴ such that (M\K, ω)and(R⁴\Ω, ω0)are symplectomorphic. Then(M, ω)is symplectomorphic to (R⁴, ω0).

In a later lecture we will be able to prove a stronger version of this statement, as a corollary of some classification results for symplectic fillings of contact manifolds (cf. Theorem 1.57).

Another interesting question is the following: suppose (M1, ω1) and (M2, ω2) are symplectic manifolds of the same dimension 2n, possibly with boundary, such that there exists a smooth embeddingM1 ֒→M2. Can one also find asymplectic embedding (M1, ω1) ֒→ (M2, ω2)? What phenomena related to the symplectic structures can prevent this? There’s one obstruction that jumps out immediately: there can be no such embedding unless

Z

M₁

ωⁿ₁ ≤ Z

M₂

ωⁿ₂,

i.e. M1 has no more volume thanM2. In dimension two there’s nothing more to say, because symplectic and area preserving maps are the same thing. But in dimension 2n for n ≥2, it was not known for a long time whether there are obstructions

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to symplectic embeddings other than the volume. A good thought experiment along these lines is the “squeezing” question: denote byB_r²ⁿ the ball of radius rabout the origin in R²ⁿ. Then it’s fairly obvious that for any r, R > 0 one can always find a volume preserving embedding

B_r²ⁿ ֒→B_R² ×R²ⁿ⁻²,

even ifr > R, for then one can “squeeze” the first two dimensions ofB_r²ⁿintoB_R² but make up for it by spreading out further inR²ⁿ⁻². But can one do thissymplectically? The answer was provided by the following groundbreaking result:

Theorem 1.24 (Gromov’s “non-squeezing” theorem [Gro85]). There exists a symplectic embedding of (B_r²ⁿ, ω0) into (B_R² ×R²ⁿ⁻², ω0) if and only if r ≤R.

This theorem was one of the first important applications of pseudoholomorphic curves, and we will spend a great deal of time in the following lectures learning the technical machinery that is needed to understand the proof.

We will close this brief introduction to symplectic topology by sketching the proof of a result that was introduced in [Gro85] and later generalized by McDuff, and provides us with a good excuse to introduce J-holomorphic curves. Recall from

§3that CP² admits a singular foliation by embedded spheres that all intersect each other at one point, and all can be parametrized by holomorphic maps CP¹ →CP². One can check that these spheres are also symplectic submanifolds with respect to the standard symplectic structure ω0 introduced in Example 1.12; moreover, they intersect each other positively, so their self-intersection numbers are always 1. The following result essentially says that the existence of such a symplectically embedded sphere is a rare phenomenon: it can only occur in a very specific set of symplectic 4-manifolds, of which (CP², ω0) is the simplest. It also illustrates an important feature of symplectic topology specifically in four dimensions: once you find a single holomorphic curve with sufficiently nice local properties, it can sometimes fully determine the manifold in which it lives.

Theorem1.25 (M. Gromov [Gro85] and D. McDuff [McD90]). Suppose(M, ω)is a closed symplectic 4-manifold containing a symplectically embedded sphere C ⊂M with self-intersection C•C = 1, but no symplectically embedded sphere with self- intersection −1. Then (M, ω) is symplectomorphic to (CP², cω0), where c > 0 is a constant and ω0 is the standard symplectic form on CP².

The idea of the proof is to choose appropriate data so that the symplectic submanifold C ⊂ M can be regarded in some sense as a holomorphic curve, and then analyze the global structure of the space of holomorphic curves to which it belongs.

It turns out that for a combination of analytical and topological reasons, this space will contain a smooth family of embedded holomorphic spheres that fill all of M and all intersect each other at one point, thus reproducing the singular foliation of Figure 1. This type of decomposition is a well known object in algebraic geometry and has more recently become quite popular in symplectic topology as well: it’s called a Lefschetz pencil. As we’ll see when we generalize Theorem 1.25 in a later lecture, there is an intimate connection between isotopy classes of Lefschetz pencils and deformation classes of symplectic structures: in the present case, the existence

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of this Lefschetz pencil implies that (M, ω) is symplectically deformation equivalent to (CP², ω0), and thus also symplectomorphic due to the Moser stability theorem (see Exercise 1.22).

The truly nontrivial part of the proof is the analysis of the moduli space of holomorphic curves, and this is what we’ll concentrate on for the next several lectures.

As a point of departure, consider the formulation (1.1) of the Cauchy-Riemann equations at the beginning of this lecture. Here u was a map from an open subset of C^m into Cⁿ, but one can also make sense of (1.1) when u is a map between two complex manifolds. In such a situation, u is called holomorphic if and only if it looks holomorphic in any choice of holomorphic local coordinates. To put this in coordinate-free language, the tangent spaces of any complex manifold X are naturally complex vector spaces, on which multiplication byimakes sense, thus defining a natural bundle endomorphism

i:T X →T X

that satisfiesi² =−Id. Then (1.1) makes sense globally and is the equation defining holomorphic maps between any two complex manifolds.

In the present situation, we’re interested in smooth maps u : CP¹ → M. The domain is thus a complex manifold, but the target might not be, which means we lack an ingredient needed to write down the right hand side of (1.1). It turns out that one can always find an object to fill this role, i.e. a fiberwise linear map J :T M →T M with the following properties:

• J² =−Id,

• ω(·, J·) defines a Riemannian metric on M.

The first condition allows us to interpretJ as “multiplication byi”, thus turning the tangent spaces ofM into complex vector spaces. The second reproduces the relation between i and ω0 that exists in R²ⁿ, thus generalizing the important interaction between symplectic and complex that we illustrated in §1: complex subspaces of T M are also symplectic, and their areas can be computed in terms of ω. These conditions make J into a compatible almost complex structure on (M, ω); we will show by fairly elementary methods in Lecture 5 that such objects always exist on symplectic manifolds, in fact there are many of them. Now, the fact that C is embedded in M symplectically also allows us to arrange the following additional condition:

• the tangent spaces T C ⊂T M are invariant underJ.

We are thus ready to introduce the following generalization of the Cauchy- Riemann equation: consider smooth maps u : CP¹ → M whose differential is a complex linear map at every point, i.e.

(1.11) T u◦i=J◦T u.

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Solutions to (1.11) are calledpseudoholomorphic, or more specifically,J-holomorphic spheres in M. Now pick a point x0 ∈ C and consider the following space of J- holomorphic spheres,

M:={u∈C^∞(CP¹, M) |T u◦i=J◦T u,

u∗[CP₁] = [C]∈H₂(M), u(0) = x0}/∼,

where u ∼ u^′ if there is a holomorphic diffeomorphism ϕ : CP¹ → CP¹ such that u^′ = u◦ϕ and ϕ(0) = 0. We assign to M the natural topology defined by C^∞- convergence of smooth maps CP¹ →M.

Lemma 1.26. M is not empty: in particular it contains an embedded J-holo- morphic sphere whose image is C.

Proof. SinceChasJ-invariant tangent spaces, any diffeomorphismu0 :CP¹ → C with u0(0) =x0 allows us to pull backJ to an almost complex structurej :=u^∗₀J on CP¹. As we’ll review in Lecture 4, the uniqueness of complex structures on S² then allows us to find a diffeomorphism ϕ : CP¹ → CP¹ such that ϕ(0) = 0 and ϕ^∗j =i, thus the desired curve isu:=u0◦ϕ. ¤ The rest of the work is done by the following rather powerful lemma, which describes the global structure of M. Its proof requires a substantial volume of analytical machinery which we will develop in the coming lectures; note that since M is not a complex manifold, the methods of complex analysis play only a minor role in this machinery, and are subsumed in particular by the theory of nonlinear elliptic PDEs. This is the point where we need the technical assumptions that C •C = 1 and M contains no symplectic spheres of self-intersection −1,⁴ as such topological conditions figure into the index computations that determine the local structure of M.

Lemma 1.27. Mis compact and admits the structure of a smooth 2-dimensional manifold. Moreover, the curves in M are all embeddings that do not intersect each other except at the point x0; in particular, they foliate M \ {x0}.

By this result, the curves in Mform the fibers of a symplectic Lefschetz pencil on (M, ω), so that the latter’s diffeomorphism and symplectomorphism type are completely determined by the moduli space of holomorphic curves.

6. Contact geometry and the Weinstein conjecture

Contact geometry is often called the “odd-dimensional cousin” of symplectic geometry, and one context in which it arises naturally is in the study of Hamil- tonian dynamics. Again we shall only sketch the main ideas; the book [HZ94] is recommended for a more detailed account.

4As we’ll see, the assumption of no symplectic spheres with self-intersection−1 is a surprisingly weak one: it can always be attained by modifying (M, ω) in a standard way known as “blowing down”.

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Consider a 2n-dimensional symplectic manifold (M, ω) with a Hamiltonian H : M →R. By the definition of the Hamiltonian vector field,dH(XH) =−ω(XH, XH) = 0, thus the flow of XH preserves the level sets

Sc :=H⁻¹(c)

for c ∈ R. If c is a regular value of H then Sc is a smooth manifold of dimension 2n−1, called a regular energy surface, and XH restricts to a nowhere zero vector field on Sc.

Exercise 1.28. If Sc ⊂M is a regular energy surface, show that the direction of XH is uniquely determined by the condition ω(XH,·)|T Sc = 0.

The directions in Exercise1.28define the so-calledcharacteristic line field onSc: its existence implies that the paths traced out on Sc by orbits of XH depend only onSc and on the symplectic structure, not on H itself. In particular, a closed orbit of XH on Sc is merely a closed integral curve of the characteristic line field. It is thus meaningful to ask the following question:

Question. Given a symplectic manifold(M, ω)and a smooth hypersurfaceS ⊂ M, does the characteristic line field on S have any closed integral curves?

We shall often refer to closed integral curves of the characteristic line field on S ⊂ M simply as closed orbits on S. There are examples of Hamiltonian systems that have no closed orbits at all, cf. [HZ94, §4.5]. However, the following result (and the related result of A. Weinstein [Wei78] for convex energy surfaces) singles out a special class of hypersurfaces for which the answer is always yes:

Theorem 1.29 (P. Rabinowitz [Rab78]). Every star-shaped hypersurface in the standard symplectic R²ⁿ admits a closed orbit.

Recall that a hypersurfaceS ⊂R²ⁿis calledstar-shaped if it doesn’t intersect the origin and the projectionR²ⁿ\{0} →S²ⁿ⁻¹ :z 7→z/|z|restricts to a diffeomorphism S → S²ⁿ⁻¹ (see Figure 2). In particular, S is then transverse to the radial vector field

(1.12) Y := 1

2

n

X

i=1

µ pi

∂

∂pi

+qi

∂

∂qi

¶ .

Exercise 1.30. Show that the vector field Y of (1.12) satisfies LYω₀ =ω₀. Definition 1.31. A vector fieldY on a symplectic manifold (M, ω) is called a Liouville vector field if it satisfies LYω=ω.

By Exercise 1.30, star-shaped hypersurfaces in R²ⁿ are always transverse to a Liouville vector field, and this turns out to be a very special property.

Definition 1.32. A hypersurface S in a symplectic manifold (M, ω) is said to be of contact type if some neighborhood of S admits a Liouville vector field that is transverse to S.

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Figure 2. A star-shaped hypersurface inR².

Given a closed contact type hypersurface S ⊂ (M, ω), one can use the flow of the Liouville vector field Y to produce a very nice local picture of (M, ω) near S.

Define a 1-form on S by

α=ιYω|S, and choose ǫ >0 sufficiently small so that

Φ : (−ǫ, ǫ)×S →M : (t, x)7→ϕ^t_Y(x) is an embedding, where ϕ^t_Y denotes the flow of Y.

Exercise 1.33.

(1) Show that the flow of Y “dilates” the symplectic form, i.e. (ϕ^t_Y)^∗ω=e^tω.

(2) Show that Φ^∗ω =d(e^tα), where we define α as a 1-form on (−ǫ, ǫ)×S by pulling it back through the natural projection to S. Hint: show first that if λ :=ιYω, then Φ^∗λ=e^tα, and notice that dλ =ω by the definition of a Liouville vector field.

(3) Show that dα restricts to a nondegenerate skew-symmetric 2-form on the hyperplane field ξ := kerα over S. As a consequence, ξ is transverse to a smooth line field ℓ on S characterized by the property that X ∈ ℓ if and only ifdα(X,·) = 0.

(4) Show that on each of the hypersurfaces{c} ×S forc∈(−ǫ, ǫ), the line field ℓdefined above is the characteristic line field with respect to the symplectic formd(e^tα).

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Several interesting consequences follow from Exercise1.33. In particular, the use of a Liouville vector field to identify a neighborhood of S with (−ǫ, ǫ)×S gives us a smooth family of hypersurfaces Sc := {c} ×S whose characteristic line fields all have exactly the same dynamics. This provides some intuitive motivation to believe Theorem 1.29: it’s sufficient to find one hypersurface in the family Sc that admits a periodic orbit, for then they all do. As it turns out, one can prove a variety of

“almost existence” results in 1-parameter families of hypersurfaces, e.g. in (R²ⁿ, ω0), a result of Hofer-Zehnder [HZ90] and Struwe [Str90] implies that for any smooth 1-parameter family of hypersurfaces, almost every (in a measure theoretic sense) hypersurface in the family admits a closed orbit. This gives a proof of the following generalization of Theorem 1.29:

Theorem1.34 (C. Viterbo [Vit87]). Every contact type hypersurface in(R²ⁿ, ω0) admits a closed orbit.

Having generalized this far, it’s natural to wonder whether the crucial properties of a contact hypersurface can be considered independently of its embedding into a symplectic manifold. The answer comes from the 1-form α and hyperplane distribution ξ= kerα⊂T S in Exercise 1.33.

Definition1.35. Acontact formon a (2n−1)-dimensional manifold is a smooth 1-form α such that dα is always nondegenerate on ξ := kerα. The hyperplane distribution ξ is then called a contact structure.

Exercise1.36. Show that the condition ofdαbeing nondegenerate onξ = kerα is equivalent to α ∧ (dα)ⁿ⁻¹ being a volume form on S, and that ξ is nowhere integrable if this is satisfied.

Given an orientation of S, we call the contact structure ξ = kerα positive if the orientation induced by α∧(dα)ⁿ⁻¹ agrees with the given orientation. One can show that if S ⊂(M, ω) is a contact type hypersurface with the natural orientation induced from M and a transverse Liouville vector field, then the induced contact structure is always positive.

Note that Liouville vector fields are far from unique, in fact:

Exercise 1.37. Show that if Y is a Liouville vector field on (M, ω) and XH is any Hamiltonian vector field, then Y +XH is also a Liouville vector field.

Thus the contact formα=ιYω|S induced on a contact type hypersurface should not be considered an intrinsic property of the hypersurface. As the next result indicates, the contact structure is the more meaningful object.

Proposition 1.38. Up to isotopy, the contact structure ξ = kerα induced on a contact type hypersurface S ⊂ (M, ω) by α =ιYω|S is independent of the choice of Y.

The proof of this is a fairly easy exercise using a standard fundamental result of contact geometry:

Theorem 1.39 (Gray’s stability theorem). If S is a closed (2n−1)-dimensional manifold and {ξt}t∈[0,1] is a smooth 1-parameter family of contact structures on S,

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then there exists a smooth 1-parameter family of diffeomorphisms {ϕt}t∈[0,1] such that ϕ0 = Id and (ϕt)∗ξ0 =ξt.

This is yet another application of the Moser deformation trick; we’ll explain the proof at the end of this section. Note that the theorem provides an isotopy between any two deformation equivalent contact structures, but there is no such result for contactforms—that’s one of the reaons why contact structures are considered to be more geometrically natural objects.

By now we hopefully have sufficient motivation to study odd-dimensional manifolds with contact structures. The pair (S, ξ) is called acontact manifold, and for two contact manifolds (S1, ξ1) and (S2, ξ2) of the same dimension, a smooth embedding ϕ :S1 ֒→S2 is called a contact embedding

(S1, ξ1)֒→(S2, ξ2)

if ϕ∗ξ1 = ξ2. If ϕ is also a diffeomorphism, then we call it a contactomorphism. One of the main questions in contact topology is how to distinguish closed contact manifolds that aren’t contactomorphic. We’ll touch upon this subject in the next section.

But first there is more to say about Hamiltonian dynamics. We saw in Exer- cise 1.33that the characteristic line field on a contact type hypersurfaceS ⊂(M, ω) can be described in terms of a contact form α: it is the unique line field containing all vectors X such that dα(X,·) = 0, and is necessarily transverse to the contact structure. The latter implies that α is nonzero in this direction, so we can use it to choose a normalization, leading to the following definition.

Definition 1.40. Given a contact form α on a (2n−1)-dimensional manifold S, the Reeb vector field is the unique vector field Xα satisfying

dα(Xα,·) = 0, and α(Xα) = 1.

Thus closed integral curves on contact hypersurfaces can be identified with closed orbits of their Reeb vector fields.⁵ The “intrinsic” version of Theorems1.29and1.34 is then the following famous conjecture.

Conjecture 1.41 (Weinstein conjecture). For every closed odd-dimensional manifold M with a contact form α, Xα has a closed orbit.

The Weinstein conjecture is still open in general, though a proof in dimension three was produced recently by C. Taubes [Tau07], using Seiberg-Witten theory.

Before this, there was a long history of partial results using the theory of pseudoholomorphic curves, such as the following (see Definition1.52below for the definition of “overtwisted”):

Theorem1.42 (Hofer [Hof93]).Every Reeb vector field on a closed3-dimensional overtwisted contact manifold admits a contractible periodic orbit.

5Note that since Liouville vector fields are not unique, the Reeb vector field on a contact hypersurface is not uniquely determined, but itsdirection is.

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u

˙Σ

{+∞} ×M

{−∞} ×M

Figure 3. A three-punctured pseudoholomorphic torus in the symplectization of a contact manifold.

The key idea introduced in [Hof93] was to look at J-holomorphic curves for a suitable class of almost complex structures J in the so-called symplectization (R× M, d(e^tα)) of a manifold M with contact form α. Since the symplectic form is now exact, it’s no longer useful to consider closed holomorphic curves, e.g. a minor generalization of (1.3) shows that all J-holomorphic spheres u:CP¹ →R×M are constant:

Area(u) =kduk²_L2 = Z

CP¹

u^∗d(e^tα) = Z

∂CP¹

u^∗(e^tα) = 0.

Instead, one considers J-holomorphic maps u: ˙Σ→R×M,

where ˙Σ denotes a closed Riemann surface with finitely many punctures. It turns out that under suitable conditions, the image of u near each puncture approaches {±∞} ×M and becomes asymptotically close to a cylinder of the formR×γ, where γ is a closed orbit of Xα (see Figure 3). Thus an existence result for punctured holomorphic curves in R×M implies the Weinstein conjecture onM.

To tie up a loose end, here’s the proof of Gray’s stability theorem, followed by another important contact application of the Moser trick.

Proof of Theorem 1.39. AssumeS is a closed manifold with a smooth family of contact forms {αt}t∈[0,1] defining contact structures ξt = kerαt. We want to find a time-dependent vector field Yt whose flowϕt satisfies

(1.13) ϕ^∗_tαt =ftα0

for some (arbitrary) smooth 1-parameter family of functions ft :S → R. Differen- tiating this expression and writing ˙ft:= _∂t^∂ft and ˙αt := _∂t^∂αt, we have

ϕ^∗_t( ˙αt+LYtαt) = ˙ftα0 = f˙t

ft

ϕ^∗_tαt, and thus

(1.14) α˙t+dιYtαt+ιYtdαt =gtαt,

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where we define a new family of functions gt:S →R via the relation

(1.15) gt◦ϕt = f˙t

ft

= ∂

∂tlogft.

Now to make life a bit simpler, we assume (optimistically!) thatYtis always tangent to ξt, hence αt(Yt) = 0 and the second term in (1.14) vanishes. We therefore need to find a vector field Yt and functiongt such that

(1.16) dαt(Yt,·) =−α˙t+gtαt.

Plugging in the Reeb vector field Xαt on both sides, we find 0 =−α˙t(Xαt) +gt,

which determines the function gt. Now restricting both sides of (1.16) to ξt, there is a unique solution for Yt since dαt|ξt is nondegenerate. We can then integrate this vector field to obtain a family of diffeomorphismsϕt, and integrate (1.15) to obtain

ft so that (1.13) is satisfied. ¤

Exercise 1.43. Try to adapt the above argument to construct an isotopy such that ϕ^∗_tαt = α0 for any two deformation equivalent contact forms. But don’t try very hard.

Finally, just as there is no local symplectic geometry, there is no local contact geometry either:

Theorem 1.44 (Darboux’s theorem for contact manifolds). Near every point in a (2n+ 1)-dimensional manifold S with contact formα, there are local coordinates (p₁, . . . , pn, q₁, . . . , qn, z) in which α=dz+P

ipi dqi.

Exercise 1.45. Prove the theorem using a Moser argument. If you get stuck, see [Gei08].

7. Symplectic fillings of contact manifolds

In the previous section, contact manifolds were introduced as objects that occur naturally as hypersurfaces in symplectic manifolds. In particular, every contact manifold (M, ξ) with contact formα is obviously a contact type hypersurface in its own symplectization (R×M, d(e^tα)), though this example is in some sense trivial.

By contrast, it is far from obvious whether any given contact manifold can occur as a contact hypersurface in a closed symplectic manifold, or relatedly, if it is a

“contact type boundary” of some compact symplectic manifold.

Definition 1.46. A compact symplectic manifold (W, ω) with boundary is said to have convex boundary if there exists a Liouville vector field in a neighborhood of

∂W that points transversely out of ∂W.

Definition 1.47. A strong symplectic filling (also called a convex filling) of a closed contact manifold (M, ξ) is a compact symplectic manifold (W, ω) with convex boundary, such that ∂W with the contact structure induced by a Liouville vector field is contactomorphic to (M, ξ).

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Since we’re now considering symplectic manifolds that are not closed, it’s also possible for ω to be exact. Observe that a primitive λ of ω always gives rise to a Liouville vector field, since the unique vector field Y defined by ιYω = λ then satisfies

LYω=dιYω =dλ =ω.

Definition 1.48. A strong filling (W, ω) of (M, ξ) is called an exact filling if ω = dλ for some 1-form λ such that the vector field Y defined by ιYω = λ points transversely out of ∂W.

Exercise 1.49. Show that if (W, ω) is a compact symplectic manifold with boundary, Y is a Liouville vector field defined near ∂W and λ = ιYω, then Y is positively transverse to ∂W if and only if λ|∂W is a positive contact form.

The exercise makes possible the following alternative formulations of the above definitions:

(1) A compact symplectic manifold (W, ω) with boundary is a strong filling if

∂W admits a contact form that extends to a primitive of ω on a neighborhood of ∂W.

(2) A strong filling is exact if the primitive mentioned above can be extended globally overW.

(3) A strong filling is exact if it has a transverse outward pointing Liouville vector field near ∂W that can be extended globally over W.

By now you’re surely wondering what a “weak” filling is. Observe that for any strong filling (W, ω) with Liouville vector field Y and induced contact structure ξ= kerιYω on the boundary,ω has a nondegenerate restriction toξ (see Exercise1.33).

The latter condition can be expressed without mentioning a Liouville vector field, hence:

Definition 1.50. A weak symplectic filling of a closed contact manifold (M, ξ) is a compact symplectic manifold (W, ω) with boundary, such that there exists a diffeomorphism ϕ:∂W →M and ω has a nondegenerate restriction toϕ^∗ξ.

Remark 1.51. One important definition that we are leaving out of the present discussion is that of a Stein filling: this is a certain type of complex manifold with contact boundary, which is also an exact symplectic filling. The results we’ll prove in these lectures for strong and exact fillings apply to Stein fillings as well, but we will usually not make specific mention of this since the Stein condition itself has no impact on our general setup. Much more on Stein manifolds can be found in the monographs [OS04] and [CE].

A contact manifold is called exactly/strongly/weakly fillable if it admits an exact/strong/weak filling. Recall that in the smooth category, every 3-manifold is the boundary of some 4-manifold; by contrast, we will see that many contact 3-manifolds are not symplectically fillable.

The unit ball in (R⁴, ω0) obviously has convex boundary: the contact structure induced onS³ is called thestandard contact structureξ0. But there are other contact structures on S³ not contactomorphic to ξ0, and one way to see this is to show that

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θ

ρ

Figure 4. An overtwisted contact structure.

they are not fillable. Indeed, it is easy (via “Lutz twists”, see [Gei08] or [Gei06]) to produce a contact structure on S³ that is overtwisted. Note that the following is not the standard definition⁶ of this term, but is equivalent due to a deep result of Eliashberg [Eli89].

Definition 1.52. A contact 3-manifold (M, ξ) is overtwisted if it admits a contact embedding of (S¹×D, ξOT), where D⊂R² is the closed unit disk and ξOT is a contact structure of the form

ξOT = ker [f(ρ)dθ+g(ρ)dφ]

with θ ∈ S¹, (ρ, φ) denoting polar coordinates on D, and (f, g) : [0,1] → R² \ {0}

a smooth path that begins at (1,0) and winds counterclockwise around the origin, making at least one half turn.

For visualization, a portion of the domain (S¹ ×D, ξOT) is shown in Figure 4.

One of the earliest applications of holomorphic curves in contact topology was the following nonfillability result.

Theorem 1.53 (M. Gromov [Gro85] and Ya. Eliashberg [Eli90]). If (M, ξ) is closed and overtwisted, then it is not weakly fillable.

The Gromov-Eliashberg proof worked by assuming a weak filling (W, ω) of (M, ξ) exists, then constructing a family ofJ-holomorphic disks inW with boundaries on a totally real submanifold in M and showing that this family leads to a contradiction if (M, ξ) contains an overtwisted disk. We will later present a proof that is similar in spirit but uses slightly different techniques: instead of dealing with boundary

6It is standard to call a contact 3-manifold (M, ξ) overtwisted if it contains an embedded overtwisted disk, which is a diskD ⊂M such thatT(∂D)⊂ξbutTD|∂D 6=ξ|∂D.

Lectures on Holomorphic Curves in Symplectic and Contact Geometry